Institute for Geometry and Physics

We give explicit expressions for the finite frequency greybody factor, quasinormal modes, and Love numbers of Kerr black holes by computing the exact connection coefficients of the radial and angular parts of the Teukolsky equation. This is obtained by solving the connection problem of the confluent Heun equation in terms of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the Alday, Gaiotto, and Tachikawa correspondence. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein equation in other interesting gravitational backgrounds.

We present a novel study of Kerr Compton amplitudes in a partial wave basis in terms of the Nekrasov-Shatashvili (NS) function of the confluent Heun equation (CHE). Remarkably, NS-functions enjoy analytic properties and symmetries that are naturally inherited by the Compton amplitudes. Based on this, we characterize the analytic dependence of the Compton phase shift in the Kerr spin parameter and provide a direct comparison to the standard post-Minkowskian (PM) perturbative approach within general relativity (GR). We also analyze the universal large frequency behavior of the relevant characteristic exponent of the CHE—also known as the renormalized angular momentum—and find agreement with numerical computations. Moreover, we discuss the analytic continuation in the harmonics quantum number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo>ℓ</a:mo></a:math> of the partial wave, and show that the limit to the physical integer values commutes with the PM expansion of the observables. Finally, we obtain the contributions to the tree-level, point-particle, gravitational Compton amplitude in a covariant basis through <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi mathvariant="script">O</c:mi><c:mo stretchy="false">(</c:mo><c:msubsup><c:mi>a</c:mi><c:mi>BH</c:mi><c:mn>8</c:mn></c:msubsup><c:mo stretchy="false">)</c:mo></c:math>, without the need to take the superextremal limit for Kerr spin. Published by the American Physical Society 2024

Abstract We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU (2) pure super Yang–Mills and $$N_f=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> on a circle.

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ordinary differential equations with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasinormal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulas for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover, we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the Denef-Hartnoll-Sachdev formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulas can be given for higher spin matter and radiation fields in more general gravitational backgrounds. Published by the American Physical Society 2024

We show that the nonperturbative dynamics of N=2 super-Yang-Mills theories in a self-dual Ω background and with arbitrary simple gauge group is fully determined by studying renormalization group equations of vacuum expectation values of surface operators generating one-form symmetries. The corresponding system of equations is a nonautonomous Toda chain, the time being the renormalization group scale. We obtain new recurrence relations which provide a systematic algorithm computing multi-instanton corrections from the tree-level one-loop prepotential as the asymptotic boundary condition of the renormalization group equations. We exemplify by computing the E_{6} and G_{2} cases up to two instantons.

Abstract We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $$GL(N,{\mathbb {C}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be written in terms of a Fredholm determinant of Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $$SL(2,{\mathbb {C}})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> this combinatorial expression takes the form of a dual Nekrasov–Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results we also propose a definition of the tau function of the Riemann–Hilbert problem on a torus with generic jump on the A-cycle.

We propose a method to perform an exact calculation of one-loop quantum corrections to black hole entropy in terms of Virasoro semiclassical blocks. We analyze in detail a four-dimensional Kerr black hole and show that in the near-extremal limit a branch of long-lived modes arises. We prove that the contribution of these modes accounts for a (s-1/2)logT_{Hawking} correction to the entropy for massless particles of spin s=1, 2. We show that in the full calculation performed in the exact Kerr background the leading contribution actually is sourced by the near-horizon region only, and as such has a universal validity for any asymptotic behavior at infinity.

We study covariant derivatives on a class of centered bimodules [Formula: see text] over an algebra [Formula: see text] We begin by identifying a [Formula: see text]-submodule [Formula: see text] which can be viewed as the analogue of vector fields in this context; [Formula: see text] is proven to be a Lie algebra. Connections on [Formula: see text] are in one-to-one correspondence with covariant derivatives on [Formula: see text]. We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived.

Abstract In this note, we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on U ( N ) $${\mathcal N}=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> gauge theories in four dimensions with matter in the adjoint and in the fundamental representations of the gauge group, respectively, and find rigorous lower bounds for the convergence radius in the two cases: if the theory is conformal , then the series has at least a finite radius of convergence, while if it is asymptotically free it has infinite radius of convergence. Via AGT correspondence, this implies that the related irregular conformal blocks of $$W_N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> algebrae admit a power expansion in the modulus converging in the whole plane. By specifying to the SU (2) case, we apply our results to analyze the convergence properties of the corresponding Painlevé $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -functions.

We study the relation between class S theories on punctured tori and\nisomonodromic deformations of flat SL(N) connections on the two dimensional\ntorus with punctures. Turning on the self dual $\\Omega$-background corresponds\nto a deautonomization of the Seiberg-Witten integrable system which implies a\nspecific time dependence in its Hamiltonians. We show that the corresponding\n$\\tau$-function is proportional to the dual gauge theory partition function,\nthe proportionality factor being a non trivial function of the solution of the\ndeautonomized Seiberg-Witten integrable system. This is obtained by mapping the\nisomonodromic deformation problem to $W_N$ free fermion correlators on the\ntorus.\n

Abstract The partition function of super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual Ω background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a non-autonomous Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical solution is obtained just by fixing the perturbative part of the IR prepotential as its asymptotic boundary condition for the RGE. We analyze the explicit solutions of the τ -system for all the classical groups at the diverse levels, extend our analysis to affine twisted Lie algebras and provide conjectural bilinear relations for the τ -functions of linear quiver gauge theory.

We study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

Abstract We introduce Riemann–Hilbert problems determined by the refined Donaldson–Thomas theory. They involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a quantum torus algebra. We study the simplest case in detail and use the Barnes double gamma function to construct a solution.

A bstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A -type by decorated Newton polygons.

Abstract In this paper, we analyze the relevance of the generalized Kronheimer construction for the gauge/gravity correspondence. We begin with the general structure of D3-brane solutions of type IIB supergravity on smooth manifolds $$Y^\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mi>Γ</mml:mi> </mml:msup> </mml:math> that are supposed to be the crepant resolution of quotient singularities $$\mathbb {C}^3/\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> with $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> a finite subgroup of SU (3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms $$\omega ^{2,1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Although excluded in the classical Kronheimer construction, they may be reintroduced by means of mass deformations. Next we concentrate on the other essential item for the D3-brane construction, namely, the existence of a Ricci-flat metric on $$Y^\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mi>Γ</mml:mi> </mml:msup> </mml:math> . We study the issue of Ricci-flat Kähler metrics on such resolutions $$Y^\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mi>Γ</mml:mi> </mml:msup> </mml:math> , with particular attention to the case $$\Gamma =\mathbb {Z}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:math> . We advance the conjecture that on the exceptional divisor of $$Y^\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Y</mml:mi> <mml:mi>Γ</mml:mi> </mml:msup> </mml:math> the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on $$\mathrm{tot} K_{{\mathbb {W}P}[112]}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>tot</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:mo>[</mml:mo> <mml:mn>112</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> that we construct, i.e., the total space of the canonical bundle of the weighted projective space $${\mathbb {W}P}[112]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:mo>[</mml:mo> <mml:mn>112</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , which is a partial resolution of $$\mathbb {C}^3/\mathbb {Z}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:math> . For the full resolution, we have $$Y^{\mathbb {Z}_4}={\text {tot}} K_{\mathbb {F}_{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>Y</mml:mi> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:msup> <mml:mo>=</mml:mo> <mml:mtext>tot</mml:mtext> <mml:msub> <mml:mi>K</mml:mi> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:msub> </mml:mrow> </mml:math> , where $$\mathbb {F}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on $$\mathbb {F}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> produced by the Kronheimer construction as initial datum in a Monge–Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential but with two different choices of the variables. In both cases, one can establish a series solution in powers of the variable along the fibers of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, rather it uniquely determines all the subsequent terms as local functionals of this initial datum. Although a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation, we have identified some new properties of this type of MA equations that we believe to be so far unknown.

By using Monge–Ampère geometry, we study the singular structure of a class of nonlinear Monge–Ampère equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge–Ampère geometry by examining the role of an induced metric on Lagrangian submanifolds of the cotangent bundle. In particular, we show that the signature of the metric serves as a classification of the Monge–Ampère equation, while singularities and elliptic–hyperbolic transitions are revealed by degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations.

A bstract We notice that the famous pentagon identity for quantum dilogarithm functions and the five-term relation for certain operators related to Macdonald polynomials discovered by Garsia and Mellit can both be understood as specific cases of a general “master pentagon identity” for group-like elements in the Ding-Iohara-Miki (or quantum toroidal, or elliptic Hall) algebra. We prove this curious identity and discuss its implications.

Abstract We explore nonequilibrium features of certain operator algebras which appear in quantum gravity. The algebra of observables in a black hole background is a Type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mrow><mml:mi>II</mml:mi></mml:mrow><mml:mi mathvariant="normal">∞</mml:mi></mml:msub></mml:mrow></mml:math> von Neumann algebra. We discuss how this algebra can be coupled to the algebra of observable of an infinite reservoir within the canonical ensemble, aiming to induce nonequilibrium dynamics. The resulting dynamics can lead the system towards a nonequilibrium steady state which can be characterized through modular theory. Within this framework we address the definition of entropy production and its relationship to relative entropy, alongside exploring other applications.

Abstract For a quasi-smooth hypersurface X in a projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>Σ</mml:mi> </mml:msub> </mml:math> , the morphism $$i^*:H^p(\mathbb {P}_{\Sigma })\rightarrow H^p(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>i</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>Σ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> induced by the inclusion is injective for $$p=\dim X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mo>dim</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> and an isomorphism for $$p&lt;\dim X-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mo>dim</mml:mo> <mml:mi>X</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . This allows one to define the Noether–Lefschetz locus $$\mathrm{NL}_{\beta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>NL</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> as the locus of quasi-smooth hypersurfaces of degree $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> such that $$i^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>i</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if $$\dim \mathbb {P}_{\Sigma }=2k+1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>dim</mml:mo> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>Σ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$k\beta -\beta _0=n\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mi>β</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>β</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mi>η</mml:mi> </mml:mrow> </mml:math> , $$n\in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , where $$\eta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>η</mml:mi> </mml:math> is the class of a 0-regular ample divisor, and $$\beta _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>β</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> satisfies the bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>codim</mml:mi> <mml:mspace /> <mml:mi>Z</mml:mi> <mml:mo>⩽</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .